On continuity and differentiability of the maximum values of functions

In this paper we consider functions, which are the maximal values of continuous functions on the families of compact subsets. Such functions are used, for example, in studying the geometric structure of various equilibrium surfaces: minimal surfaces, surfaces of a constant mean curvature, and so forth. Usually, such functions are constructed as the geometric characteristics of the surfaces under study, for instance, as the distance from a point of the surface to a fixed line, as the radius of the circumscribed sphere. One of the key points of this approach is the justification of their continuity and differentiability. This allows us to derive differential relations for the considered functions. In the present paper, the questions of continuity and differentiability are considered in a more general formulation, for topological and metric spaces. In particular, we find the conditions for the mapping of topological spaces $F: X \to T$ ensuring that a function of the form $\rho(t) = \max_{x \in F^{-1}(t)} g(x)$ is continuous. In addition, for such functions we obtain the conditions guaranteeing that they are Lipschitz and $\delta$-convex in $\mathbb{R}^m$.